| import Mathlib |
| import OAI.Combinatorics.MatchingCount.Sampling.MarkBridge |
|
|
| namespace OAI |
|
|
| namespace MatchingFPRAS.RationalSource |
| open Finset Transport Variation Enlargement |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _ |
|
|
| def rawReweight (w w₁ : V → V → ℚ) (K : ℕ) (e : SourceRealEdge V K) : ℚ := |
| if e.2.val=2*(2*K+2) then BottleneckAlgorithm.edge w₁ (source e.1) (target e.1)/ |
| BottleneckAlgorithm.edge w (source e.1) (target e.1) else 1 |
|
|
| def bagReweight (w w₁ : V → V → ℚ) (K : ℕ) (e : BagEdge (bags w K)) : ℚ := |
| 1+∑ q : SourceRealEdge V K,if coloredReal w K q=e then rawReweight w w₁ K q-1 else 0 |
| lemma bagReweight_on (w w₁ : V → V → ℚ) (K : ℕ) (q : SourceRealEdge V K) : |
| bagReweight w w₁ K (coloredReal w K q)=rawReweight w w₁ K q := by |
| simp only [bagReweight,(coloredReal_injective w K).eq_iff,sum_ite_eq',mem_univ,ite_true] |
| ring |
| lemma bagReweight_off (w w₁ : V → V → ℚ) (K : ℕ) (e : BagEdge (bags w K)) |
| (he : ¬∃ q,coloredReal w K q=e) : bagReweight w w₁ K e=1 := by |
| have hz : (∑ q : SourceRealEdge V K,if coloredReal w K q=e then rawReweight w w₁ K q-1 else 0)=0 := by |
| apply sum_eq_zero |
| intro q _ |
| exact ite_eq_right (fun h=>he ⟨q,h⟩) |
| rw [bagReweight,hz,add_zero] |
| omit [Fintype V] [Nontrivial V] in |
| lemma rawReweight_cast (w w₁ : V → V → ℚ) (K : ℕ) (q : SourceRealEdge V K) : |
| (rawReweight w w₁ K q:ℝ)=Enlargement.rawReweight (fun x y=>(w x y:ℝ)) |
| (fun x y=>(w₁ x y:ℝ)) K q := by |
| simp only [rawReweight,Enlargement.rawReweight] |
| split_ifs |
| · simp only [Rat.cast_div,BottleneckAlgorithm.edge,logicalActivity,Switching.edgeActivity] |
| split_ifs <;> rfl |
| · norm_cast |
| lemma bagReweight_cast (w w₁ : V → V → ℚ) (K : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) : |
| (bagReweight w w₁ K (sourceEdgeEquiv w K e):ℝ)= |
| Enlargement.bagReweight (fun x y=>(w x y:ℝ)) (fun x y=>(w₁ x y:ℝ)) K e := by |
| classical |
| by_cases he : ∃ q,sourceColoredReal (fun x y=>(w x y:ℝ)) K q=e |
| · obtain ⟨q,rfl⟩:=he |
| rw [sourceEdgeEquiv_real,bagReweight_on] |
| exact (rawReweight_cast w w₁ K q).trans (Enlargement.bagReweight_on _ _ K q).symm |
| · rw [bagReweight_off w w₁ K _ ?_] |
| · unfold Enlargement.bagReweight |
| change ¬∃ q,sourceRealEmbedding (fun x y=>(w x y:ℝ)) K q=e at he |
| rw [dite_eq_right he] |
| norm_cast |
| · rintro ⟨q,hq⟩ |
| apply he |
| refine ⟨q,(sourceEdgeEquiv w K).injective ?_⟩ |
| rwa [sourceEdgeEquiv_real] |
|
|
| def observableA (w : V → V → ℚ) (K : ℕ) (a : Marked w K) : ℚ := |
| if a.2=(fun _=>0) then 1 else 0 |
|
|
| def observableD (w w₁ : V → V → ℚ) (K : ℕ) (a : Marked w K) : ℚ := |
| if a.2=(fun _=>0) then ∏ e ∈ a.1.val,bagReweight w w₁ K e else 0 |
|
|
| def observableProbe (w w₁ : V → V → ℚ) (K : ℕ) (e₀ : LogicalEdge V) (a : Marked w K) : ℚ := |
| if {centralEdge w K e₀}⊆a.1.val ∧ |
| a.2=(fun e=>if e.val∈({centralEdge w K e₀}:Finset _) then 2 else 0) |
| then ∏ e ∈ a.1.val \ {centralEdge w K e₀},bagReweight w w₁ K e else 0 |
| end MatchingFPRAS.RationalSource |
| namespace MatchingFPRAS.Transport |
| open Finset |
|
|
| variable {V E F I : Type*} [Fintype V] [DecidableEq V] |
| variable [Fintype E] [DecidableEq E] [Fintype F] [DecidableEq F] [Fintype I] |
| variable {inc : E → V → Prop} {j : F → V → Prop} |
| omit [DecidableEq V] [Fintype E] [Fintype F] [Fintype I] in |
| lemma markedEquiv_marks_eq (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (a : Marking.Marked (I:=I) inc) (r : F → I) : |
| (markedEquiv e he a).2=(fun x=>r x.val) ↔ a.2=(fun x=>r (e x.val)) := by |
| constructor |
| · intro h |
| funext x |
| have hh:=congrFun h (selectedEquiv e he a.1 x) |
| rw [markedEquiv_mark] at hh |
| exact hh |
| · intro h |
| funext x |
| obtain ⟨y,rfl⟩:=(selectedEquiv e he a.1).surjective x |
| rw [markedEquiv_mark,congrFun h] |
| rfl |
| omit [DecidableEq V] [Fintype E] [Fintype F] in |
| lemma markedEquiv_zero (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (a : Marking.Marked (I:=Fin 3) inc) : |
| (markedEquiv e he a).2=(fun _=>0) ↔ a.2=(fun _=>0) := |
| markedEquiv_marks_eq e he a (fun _=>0) |
| omit [DecidableEq V] [Fintype E] [Fintype F] in |
| lemma markedEquiv_probe (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) |
| (a : Marking.Marked (I:=Fin 3) inc) (u : E) : |
| ({e u}⊆(markedEquiv e he a).1.val ∧ |
| (markedEquiv e he a).2=(fun x=>if x.val∈({e u}:Finset _) then 2 else 0)) ↔ |
| ({u}⊆a.1.val ∧ a.2=(fun x=>if x.val∈({u}:Finset _) then 2 else 0)) := by |
| rw [markedEquiv_marks_eq e he a (fun x=>if x∈({e u}:Finset F) then 2 else 0)] |
| rw [singleton_subset_iff,singleton_subset_iff] |
| change (e u∈a.1.val.image e ∧ _) ↔ (u∈a.1.val ∧ _) |
| simp only [mem_singleton,e.injective.eq_iff, |
| mem_image,exists_eq_right] |
| omit [DecidableEq V] [Fintype E] [Fintype F] in |
| lemma pmEquiv_product_erase {R : Type*} [CommMonoid R] |
| (e : E ≃ F) (he : ∀ a v,j (e a) v ↔ inc a v) (M : PM inc) (u : E) (f : F → R) : |
| (∏ x ∈ (pmEquiv e he M).val \ {e u},f x)=∏ x ∈ M.val \ {u},f (e x) := by |
| have hh : (pmEquiv e he M).val \ {e u}=(M.val \ {u}).image e := by |
| change M.val.image e \ {e u}=(M.val \ {u}).image e |
| ext x |
| simp only [mem_sdiff,mem_image,mem_singleton] |
| constructor |
| · rintro ⟨⟨y,hy,rfl⟩,hu⟩ |
| exact ⟨y,⟨hy,fun h=>hu (congrArg e h)⟩,rfl⟩ |
| · rintro ⟨y,⟨hy,hu⟩,rfl⟩ |
| exact ⟨⟨y,hy,rfl⟩,fun h=>hu (e.injective h)⟩ |
| rw [hh] |
| exact prod_image (fun _ _ _ _ h=>e.injective h) |
| end MatchingFPRAS.Transport |
| namespace MatchingFPRAS.RationalSource |
| open Finset Transport Variation Enlargement |
|
|
| variable {V : Type*} [Fintype V] [LinearOrder V] [Nontrivial V] |
| attribute [local instance] NumberedHierarchy.labelOrder sym2Order bagEdgeOrder bagEdgeFintype |
| attribute [local instance] UnitRefresh.Computable.allowedDec prodOrder sumOrder |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| DecidableEq (SourceLabel (fun x y=>(w x y:ℝ)) K) := Classical.decEq _ |
| noncomputable local instance (w : V → V → ℚ) (K : ℕ) : |
| Fintype (BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) := Fintype.ofFinite _ |
| lemma sourceEdgeEquiv_incidence (w : V → V → ℚ) (K : ℕ) |
| (e : BagEdge (sourceBags (fun x y=>(w x y:ℝ)) K)) (x : SourceVertex V K) : |
| (sourceEdgeEquiv w K e).Incident x ↔ e.Incident x := Iff.rfl |
| lemma sourceMarkedEquiv_zero (w : V → V → ℚ) (K : ℕ) |
| (a : SourceMarked (fun x y=>(w x y:ℝ)) K) : |
| (sourceMarkedEquiv w K a).2=(fun _=>0) ↔ a.2=(fun _=>0) := |
| markedEquiv_zero (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K) a |
| lemma observableA_cast (w : V → V → ℚ) (K : ℕ) |
| (a : SourceMarked (fun x y=>(w x y:ℝ)) K) : |
| (observableA w K (sourceMarkedEquiv w K a):ℝ)=Enlargement.observableA (fun x y=>(w x y:ℝ)) K a := by |
| simp only [observableA,Enlargement.observableA,sourceMarkedEquiv_zero] |
| split_ifs <;> norm_cast |
| lemma observableD_cast (w w₁ : V → V → ℚ) (K : ℕ) |
| (a : SourceMarked (fun x y=>(w x y:ℝ)) K) : |
| (observableD w w₁ K (sourceMarkedEquiv w K a):ℝ)= |
| Enlargement.observableD (fun x y=>(w x y:ℝ)) (fun x y=>(w₁ x y:ℝ)) K a := by |
| simp only [observableD,Enlargement.observableD,sourceMarkedEquiv_zero] |
| split_ifs |
| · push_cast |
| change matchingWeight (fun e=>(bagReweight w w₁ K e:ℝ)) |
| (pmEquiv (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K) a.1)=_ |
| rw [matchingWeight_pmEquiv (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K)] |
| unfold matchingWeight |
| apply prod_congr rfl |
| intro e _ |
| exact bagReweight_cast w w₁ K e |
| · norm_cast |
| lemma observableProbe_cast (w w₁ : V → V → ℚ) (K : ℕ) (u : LogicalEdge V) |
| (a : SourceMarked (fun x y=>(w x y:ℝ)) K) : |
| (observableProbe w w₁ K u (sourceMarkedEquiv w K a):ℝ)= |
| Enlargement.observableProbe (fun x y=>(w x y:ℝ)) (fun x y=>(w₁ x y:ℝ)) K u a := by |
| have hh:=markedEquiv_probe (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K) a |
| (Enlargement.centralEdge (fun x y=>(w x y:ℝ)) K u) |
| rw [sourceEdgeEquiv_central] at hh |
| change ({centralEdge w K u}⊆(sourceMarkedEquiv w K a).1.val ∧ |
| (sourceMarkedEquiv w K a).2=(fun e=>if e.val∈({centralEdge w K u}:Finset _) then 2 else 0)) ↔ _ at hh |
| simp only [observableProbe,Enlargement.observableProbe,hh] |
| simp only [mem_singleton] at hh ⊢ |
| split_ifs <;> try contradiction |
| · push_cast |
| rw [←sourceEdgeEquiv_central w K u] |
| change (∏ e ∈ (pmEquiv (inc:=BagEdge.Incident (bags:=sourceBags (fun x y=>(w x y:ℝ)) K)) |
| (j:=BagEdge.Incident (bags:=bags w K)) (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K) a.1).val \ |
| {sourceEdgeEquiv w K (Enlargement.centralEdge (fun x y=>(w x y:ℝ)) K u)}, |
| (bagReweight w w₁ K e:ℝ))=_ |
| rw [pmEquiv_product_erase (sourceEdgeEquiv w K) (sourceEdgeEquiv_incidence w K)] |
| refine prod_congr (by ext; simp) ?_ |
| intro e _ |
| exact bagReweight_cast w w₁ K e |
| · norm_cast |
| end MatchingFPRAS.RationalSource |
|
|
| end OAI |
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